A natural question is to determine the finiteness properties of these extensions. In [1]} it was shown that the braided Thompson's group is of type \(F_\infty \) . In stark contrast, asymptotic mapping class groups of surfaces are only known to be finitely presented in the finite genus case [2]}, [3]}, and finitely generated for infinite-genus surfaces [4]}. The following question is implicit in [2]}, [4]}, and appears explicitly in [7]} and [8]}:
